Properties of toposes of right $M$-sets are studied, and these toposes are characterised up to equivalence by their canonical points. The solution to the corresponding Morita equivalence problem is… Expand

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… Expand

We give an example of an essential, hyperconnected, local geometric morphism that is not locally connected, arising from our work-in-progress on geometric morphisms $\mathbf{PSh}(M) <- PSh(N)$, where $M$ and $N$ are monoids.Expand

FitzGerald identified four conditions (RI), (UR), (RI*) and (UR*) that are necessarily satisfied by an algebra, if its monoid of endomorphisms has commuting idempotents. We show that these conditions… Expand

We present and characterize the classes of Grothendieck toposes having enough supercompact objects or enough compact objects. In the process, we examine the subcategories of supercompact objects and… Expand

We systematically investigate, for a monoid $M$, how topos-theoretic properties of $\mathbf{PSh}(M)$, including the properties of being atomic, strongly compact, local, totally connected or cohesive, correspond to semigroup-theory properties of $M$.Expand