  We compute quantum character varieties of arbitrary closed surfaces with
boundaries and marked points. These are categorical invariants $\int_S\mathcal
A$ of a surface $S$, determined by the choice of a braided tensor category
$\mathcal A$, and computed via factorization homology.
  We identify the algebraic data governing marked points and boundary
components with the notion of a {\em braided module category} for $\mathcal A$,
and we describe braided module categories with a generator in terms of certain
explicit algebra homomorphisms called {\em quantum moment maps}. We then show
that the quantum character variety of a decorated surface is obtained from that
of the corresponding punctured surface as a quantum Hamiltonian reduction.
  Characters of braided $\mathcal A$-modules are objects of the torus category
$\int_{T^2}\mathcal A$. We initiate a theory of character sheaves for quantum
groups by identifying the torus integral of $\mathcal A=\operatorname{Rep_q} G$
with the category $\mathcal D_q(G/G)-\operatorname{mod}$ of equivariant quantum
$\mathcal D$-modules. When $G=GL_n$, we relate the mirabolic version of this
category to the representations of the spherical double affine Hecke algebra
(DAHA) $\mathbb{SH}_{q,t}$.
