  We introduce a diagrammatic monoidal category $\mathcal{H}eis_k(z,t)$ which
we call the quantum Heisenberg category, here, $k \in \mathbb{Z}$ is "central
charge" and $z$ and $t$ are invertible parameters. Special cases were known
before: for central charge $k=-1$ and parameters $z = q-q^{-1}$ and $t =
-z^{-1}$ our quantum Heisenberg category may be obtained from the deformed
version of Khovanov's Heisenberg category introduced by Licata and the second
author by inverting its polynomial generator, while $\mathcal{H}eis_0(z,t)$ is
the affinization of the HOMFLY-PT skein category. We also prove a basis theorem
for the morphism spaces in $\mathcal{H}eis_k(z,t)$.
