We introduce a simple diagrammatic 2-category A that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of sl_∞. We show that A is equivalent to a truncation of the Khovanov--Lauda categorified quantum group U of type A_∞, and also to a truncation of Khovanov's Heisenberg 2-category H. This equivalence is a categorification of the principal realization of the basic representation of sl_∞. As a result of the categorical equivalences

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... ribed above, certain actions of H induce actions of U, and vice versa. In particular, we obtain an explicit action of U on representations of symmetric groups. We also explicitly compute the Grothendieck group of the truncation of H. The 2-category 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.