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@Jens Hemelaer and I just got our article published in Applied Categorical Structures. It was a lot of fun to write. We were able to understand a whole bunch of topos properties much better (for ourselves, at least!) through the lens of monoids, and conversely to "understand monoids via geometry" in a whole new way. If I can get my act together with my own papers, it should be the start of a much bigger story..!
Congrats! Looking forward to reading it!
I gave a talk based on this paper yesterday with the title Using topos theory to make analogies precise, which has since been put online.
The talk is intended to be accessible to a wide mathematical audience. If you have ever been curious what the appeal of topos theory is, this is my answer.