  Given a Hamiltonian torus action on a symplectic manifold, Teleman and Fukaya
have proposed that the Fukaya category of each symplectic quotient should be
equivalent to an equivariant Fukaya category of the original manifold. We lay
out new conjectures that extend this story - in certain situations - to
singular values of the moment map. These include a proposal for how, in some
cases, we can recover the non-equivariant Fukaya category of the original
manifold starting from data on the quotient.
  To justify our conjectures we pass through the mirror and work out numerous
examples, using well-established heuristics in toric mirror symmetry. We also
discuss the algebraic and categorical structures that underlie our story.
