The quantum torus as an $\mathbb E_M$-category

Given an oriented $2$-manifold $M$, a locally constant sheaf of lattices $\Lambda$ over $M$, and a pointed morphism $q : \textsf B^2\Lambda \rightarrow \textsf B^4\mathbf C^{\times}$, we define an $\mathbb E_M$-category $\mathrm{Rep}_q(\check T)$ which we call the "quantum torus" at level $q$. We explain why this terminology is deserved and calculate the factorization homology of $\mathrm{Rep}_q(\check T)$. When $M$ arises from a global complex curve, we confirm (a version of) a conjecture of Ben-Zvi and Nadler for tori.