Comment by abelaer

That's right! This is secretly about doing calculus on posets. You can actually generalise some notions from incidence algebras to other settings, like groupoids and categories, where you can play the same games. This is something I mostly haven't looked at yet, but I think it might be fun, and some people seem to have found it useful (for example: https://arxiv.org/abs/1809.00941 and https://arxiv.org/abs/2501.06662)