  Building on the theory of quantum posets, we introduce a non-commutative
version of suplattices, i.e., complete lattices whose morphisms are
supremum-preserving maps, which form a step towards a new notion of quantum
topological spaces. We show that the theory of these quantum suplattices
resembles the classical theory: the opposite quantum poset of a quantum
suplattice is again a quantum suplattice, and quantum suplattices arise as
algebras of a non-commutative version of the monad of downward-closed subsets
of a poset. The existence of this monad is proved by introducing a
non-commutative generalization of monotone relations between quantum posets,
which form a compact closed category. Moreover, we introduce a non-commutative
generalization of Galois connections and we prove that an upper Galois adjoint
of a monotone map between quantum suplattices exists if and only if the map is
a morphism of quantum suplattices. Finally, we prove a quantum version of the
Knaster-Tarski fixpoint theorem: the quantum set of fixpoints of a monotone
endomap on a quantum suplattice form a quantum suplattice.
