Monoid Properties as Invariants of Toposes of Monoid Actions
My joint paper with Jens Hemelaer just got our article published in Applied Categorical Structures.
We were able to understand a many topos-theoretic properties much better (for ourselves, at least!) through the lens of monoids, and conversely to "understand monoids via geometry" in this new way, by comparing properties of these toposes with the same properties applied to localic toposes.
Since I'm currently working on topological monoids, this could be the start of a bigger story!