Toposes of Topological Monoid Actions
My paper on topological monoids is finally complete!
The purpose of this paper was to answer the basic questions about toposes of actions of topological monoids on sets. Its introduction gives a more detailed account, but here is an overview of the questions that were answered.
Are these categories of actions necessarily toposes? (Yes!)
What properties do they have?
Are they Grothendieck toposes; in particular, can we find natural sites for them? (Yes and yes!)
What information about the topological monoid can we recover from the topos if we are given its underlying discrete monoid? (We obtain a canonical coarsest topology on that monoid.)
What about if we are only given the forgetful functor to the topos of sets? (We obtain a representing topological monoid built from the natural endomorphisms of the forgetful functor.)
What can we say about the topological properties of the representative monoids we get out of the preceding two questions?
Can we characterize these toposes? (Yes, in terms of their points.)
Do continuous monoid homomorphisms (or semigroup homomorphisms) produce geometric morphisms between these toposes? (Yes!)
If so, can we use these to characterize Morita equivalence between these toposes? (Alas, no!)
All questions and feedback are welcomed.