The study of group actions on C*-algebras ("noncommutative dynamics") has been a central theme of C*-theory for many years now. But some C*-algebras (Cuntz-Krieger algebras, for example) don't quite fit into the framework of group actions---it's necessary to allow actions by "group-like" objects, including groupoids and inverse semigroups. In joint work with Nandor Sieben, we've recently been able to show a strong correspondence between actions of r-discrete groupoids and of inverse semigroups. I'll focus upon a "curious" (to me, anyway) partition of unity argument we had to scrape together to show an isomorphism of the associated crossed product C*-algebras. This argument has nothing to do with C*-algebras per se, and can be regarded as an answer to the question, "How much collapsing occurs when we identify functions which agree on overlaps?" Time permitting, I'll indicate how this leads to the crossed product isomorphism.