  Liftings of endofunctors on sets to endofunctors on relations are commonly
used to capture bisimulation of coalgebras. Lax versions have been used in
those cases where strict lifting fails to capture bisimilarity, as well as in
modeling other notions of simulation. This paper provides tools for defining
and manipulating lax liftings.
  As a central result, we define a notion of a lax distributive law of a
functor over the powerset monad, and show that there is an isomorphism between
the lattice of lax liftings and the lattice of lax distributive laws.
  We also study two functors in detail: (i) we show that the lifting for
monotone bisimilarity is the minimal lifting for the monotone neighbourhood
functor, and (ii) we show that the lattice of liftings for the (ordinary)
neighbourhood functor is isomorphic to P(4), the powerset of a 4-element set.
