We believe the answer to the following question, that is relevant to a joint research project with Piotr Szewczak, should be known. We would appreciate any help or pointer.
Needed definitions may be found in, e.g., Blass's chapter in the Handbook of Set Theory.


For $f\in\omega^\omega$, let $K\_f:=\{g\in \omega^\omega : g\le^\* f\}$.


Let $\mathfrak{tmp}$ be the minimal cardinality of a set $Y\subseteq \omega^\omega$ such that the set $\bigcup\_{f\in Y}K\_f$ is not meager.


Since every set $K\_f$ is meager, we have that
$\mathfrak{b}\le \mathfrak{tmp}\le \mathrm{non}(meager)$.


Is $\mathfrak{tmp}$ (provably) equal to either of these cardinals?



A classic cardinal characteristic of the continuum in disguise?
For every meager $M \subseteq \omega^{\omega}$ there exists $f\_M \in \omega^{\omega}$ such that for every increasing $f \in \omega^{\omega}$, $K\_f \subseteq M$ implies $f \leq^{\star} f\_M$. For a proof, see Theorem 2.2.2 in Bartozynski, Judah book. It follows that your invariant is the bounding number.


