  We introduce a simple diagrammatic 2-category $\mathscr{A}$ that categorifies
the image of the Fock space representation of the Heisenberg algebra and the
basic representation of $\mathfrak{sl}_\infty$. We show that $\mathscr{A}$ is
equivalent to a truncation of the Khovanov--Lauda categorified quantum group
$\mathscr{U}$ of type $A_\infty$, and also to a truncation of Khovanov's
Heisenberg 2-category $\mathscr{H}$. This equivalence is a categorification of
the principal realization of the basic representation of
$\mathfrak{sl}_\infty$.
  As a result of the categorical equivalences described above, certain actions
of $\mathscr{H}$ induce actions of $\mathscr{U}$, and vice versa. In
particular, we obtain an explicit action of $\mathscr{U}$ on representations of
symmetric groups. We also explicitly compute the Grothendieck group of the
truncation of $\mathscr{H}$.
  The 2-category $\mathscr{A}$ can be viewed as a graphical calculus describing
the functors of $i$-induction and $i$-restriction for symmetric groups,
together with the natural transformations between their compositions. The
resulting computational tool is used to give simple diagrammatic proofs of
(apparently new) representation theoretic identities.
