  We demonstrate that categories of continuous actions of topological monoids
on discrete spaces are Grothendieck toposes. We exhibit properties of these
toposes, giving a solution to the corresponding Morita-equivalence problem. We
characterize these toposes in terms of their canonical points. We identify
natural classes of representatives with good topological properties, `powder
monoids' and then `complete monoids', for the Morita-equivalence classes of
topological monoids. Finally, we show that the construction of these toposes
can be made (2-)functorial by considering geometric morphisms induced by
continuous semigroup homomorphisms.
