  We study toposes of actions of monoids on sets. We begin with ordinary
actions, producing a class of presheaf toposes which we characterize. As
groundwork for considering topological monoids, we branch out into a study of
supercompactly generated toposes (a class strictly larger than presheaf
toposes). This enables us to efficiently study and characterize toposes of
continuous actions of topological monoids on sets, where the latter are viewed
as discrete spaces. Finally, we refine this characterization into necessary and
sufficient conditions for a supercompactly generated topos to be equivalent to
a topos of this form.
