Motivated by spectral gluing patterns in the Betti Langlands program, we show that for any reductive group $G$, a parabolic subgroup $P$, and a topological surface $M$, the (enhanced) spectral Eisenstein series category of $M$ is the factorization homology over $M$ of the $\mathrm{ E}_2$-Hecke category $\mathrm{ H}_{ G, P} = \mathrm{ IndCoh}(\mathrm{ LS}_{ G, P}(D^2, S^1))$, where $\mathrm{ LS}_{ G, P}(D^2, S^1)$ denotes the moduli stack of $G$-local systems on a disk together with a $P$-reduction on the boundary circle. More generally, for any pair of stacks $\mathcal{ Y}\to \mathcal{ Z}$ satisfying some mild conditions and any map between topological spaces $N\to M$, we define $(\mathcal{ Y}, \mathcal{ Z})^{ N, M} = \mathcal{ Y}^N \times_{ \mathcal{ Z}^N} \mathcal{ Z}^M$ to be the space of maps from $M$ to $\mathcal{ Z}$ along with a lift to $\mathcal{ Y}$ of its restriction to $N$. Using the pair of pants construction, we define an $\mathrm{ E}_n$-category $\mathrm{ H}_n(\mathcal{ Y}, \mathcal{ Z}) = \mathrm{ IndCoh}_0\left(\left((\mathcal{ Y}, \mathcal{ Z})^{ S^{ n-1}, D^n}\right)^\wedge_{ \mathcal{ Y}}\right)$ and compute its factorization homology on any $d$-dimensional manifold $M$ with $d\leq n$, \[ \int_M \mathrm{ H}_n(\mathcal{ Y}, \mathcal{ Z}) \simeq \mathrm{ IndCoh}_0\left(\left((\mathcal{ Y}, \mathcal{ Z})^{ \partial (M\times D^{ n-d}), M}\right)^\wedge_{ \mathcal{ Y}^M}\right), \] where $\mathrm{ IndCoh}_0$ is the sheaf theory introduced by Arinkin--Gaitsgory and Beraldo. Our result naturally extends previous known computations of Ben-Zvi--Francis--Nadler and Beraldo.
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